Floer homology

In mathematics, Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. Some of these theories are due directly to Andreas Floer, while others are derived or inspired by his work. They are all modelled upon Morse homology on finite dimensional manifolds, extending it to the case where the relevant Morse function has finite relative indices. The differentials all count some sort of pseudoholomorphic curves.

Related Topics:
Mathematics - Homology theories - Andreas Floer - Morse homology - Pseudoholomorphic curve

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The several conjecturally equivalent Floer homologies of three-manifolds all yield three types of Homology groups that fit into an exact triangle. This is formally similar to the combinatorially defined Khovanov homology, which is known to be related by a spectral sequence to Heegard Floer homology. The three-manifold theories also come equipped with a distinguished element if the three-manifold is equipped with a contact structure (A contact structure is required to define embedded contact homology but not the others). They should also have corresponding relative invariants for four-manifolds with boundary valued in the Floer homologies of the boundaries. The last is closely related to the notion of a topological quantum field theory.

Related Topics:
Three-manifolds - Exact triangle - Khovanov homology - Spectral sequence - Three-manifold - Contact structure - Topological quantum field theory

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